The latest problem from fakalin. Took some wrong turns and hints to solve it…

Given a polynomial \(f: \mathbb{Z}\to \mathbb{Z}\) with positive integer coefficients, how many evaluations of \(f\) does it take to obtain the polynomial?

(An \(f: \mathbb{R}\to \mathbb{R}\) polynomial with real coefficients would take the number of degrees plus 1 to specify, which, if it held in this case, would render the answer unbounded. But the correct answer in this case is surprisingly small.)
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This is a popular card game of the 5-10-K class played in China, the rules of which are described in English here, but not in a generalizable way. Another version on Wikipedia makes logical sense but isn’t how I’ve seen it in practice. In any case, like for many card and board games, the rules are not described in a proper way for the new player and that’s annoying (some rules are not important or obvious, other rules are important but obscure, etc.). I remember in the help files of say, MS Hearts, they follow a four-section template of: (0) basics, (1) goal of the game, (2) playing the game, and (3) strategy; in that order, which I think is the perfect logic that should be used for describing all games. Why nobody wants to explain a relatively simple game as this except by enumeration of examples is a mystery to me, so here goes:

Basics:
The game is played with two joker-included decks of cards. 5, 10, and K are point cards worth face value, except K is worth 10 points. It is a standard 4-player partnership game, so a person and his cross is a team. It is played in multiple rounds and each round there is a dealer (hence a “dealer” team and an opposing “grabber” team). Each team keeps a “level” represented by a card rank (i.e. 2,3,…,A), and each round is associated with the dealer’s level, whose corresponding rank cards are called the level cards.
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